1. Integrating Bayesian Networks and Simpson’s Paradox in Data Mining Alex Freitas University of Kent Ken McGarry University of Sunderland
2. Outline of the Talk <ul><li>Introduction to Knowledge Discovery & Data Mining </li></ul><ul><li>Constructing Bayesian networks from data </li></ul><ul><li>Simpson’s paradox </li></ul><ul><li>Proposed method for integrating Bayesian networks and Simpson’s paradox </li></ul><ul><li>Conclusions </li></ul>
3. Introduction <ul><li>Data Mining consists of extracting patterns from data, and it is the core step of a knowledge discovery process </li></ul><ul><li> pre-proc data mining post-proc </li></ul><ul><li>Data interesting </li></ul><ul><li> 22, M, 30K patterns </li></ul><ul><li>26, F, 55K IF (salary = high) </li></ul><ul><li>……… . THEN (credit = good) </li></ul>
4. The Knowledge Discovery Process – a popular definition <ul><li>“ Knowledge Discovery in Databases is the non-trivial process of identifying valid, novel, potentially useful, and ultimately understandable patterns in data” </li></ul><ul><li>(Fayyad et al. 1996) </li></ul><ul><li>Focus on the quality of discovered patterns </li></ul><ul><ul><li>independent of the data mining algorithm </li></ul></ul><ul><li>This definition is often quoted, but not very seriously taken into account </li></ul><ul><ul><li>A lot of research on discovering valid, accurate patterns </li></ul></ul><ul><ul><li>Little research on discovering potentially useful patterns </li></ul></ul>
5. Criteria to Evaluate the “Interestingness” of Discovered Patterns useful novel, surprising comprehensible valid (accurate) Amount of Research Difficulty of measurement
6. On the difficulty of discovering surprising patterns in data mining <ul><li>Focus on maximizing accuracy leads to very accurate but useless rules, e.g. (Brin et al. 1997) – census data: </li></ul><ul><ul><li>IF (person is pregnant) THEN (gender is female) </li></ul></ul><ul><ul><li>IF (age 5) THEN (employed = no) </li></ul></ul><ul><li>(Tsumoto 2000) extracted 29,050 rules from a medical dataset. Out of these, just 220 (less than 1%) were considered interesting or surprising to the user </li></ul>
7. Bayesian network example A B C D A Bayesian network represents potentially causal patterns, which tend to be more useful for intelligent decision making Motivation for Integrating Bayesian Networks and Simpson’s Paradox However, algorithms for constructing Bayesian networks from data were not designed to discover surprising patterns Simpson’s paradox is surprising by nature Causality + Surprisingness tends to improve Usefulness
8. Constructing Bayesian Networks from Data <ul><li>Methods based on conditional independence tests </li></ul><ul><ul><li>Not scalable to datasets with many variables (attributes) </li></ul></ul><ul><li>Methods based on search guided by a scoring function </li></ul><ul><ul><li>Iteratively create candidate solutions (Bayesian networks) and evaluate the quality of each created network using a scoring function, until a stopping criteria is satisfied </li></ul></ul><ul><ul><li>Sequential methods consider a single candidate solution at a time </li></ul></ul><ul><ul><li>Population-based methods consider many candidate solutions at a time </li></ul></ul>
9. <ul><li>Examples of sequential method </li></ul><ul><ul><li>B algorithm starts with an empty network and at each iteration adds, to the current candidate solution, the edge that maximizes the value of the scoring function </li></ul></ul><ul><ul><li>K2 algorithm requires that the variables be ordered and the user specifies a parameter: the maximum number of parents of each variable in the network to be constructed </li></ul></ul><ul><li>Both are greedy methods (local search), which offer no guarantee of finding the optimal network </li></ul><ul><li>Population-based methods are global search methods, but are stochastic, so again no guarantees </li></ul>
10. Limitations of methods for constructing Bayesian networks from data (1) <ul><li>Theoretical limitation (best possible algorithm & data) </li></ul><ul><li>Bayesian networks are Independence maps (I-maps) of the true probability distribution </li></ul><ul><ul><li>Every independence between variables represented in the network is an actual independence in the true probability distribution </li></ul></ul><ul><ul><li>Dependences between variables represented in the network are not guaranteed to be actual dependences in the true probability distribution </li></ul></ul>
11. Limitations of methods for constructing Bayesian networks from data (2) <ul><li>Practical limitations </li></ul><ul><li>The problem of constructing the optimal net is too complex in large datasets, so we have to use methods which do not guarantee the discovery of the optimal net </li></ul><ul><li>Sampling variation and/or noisy data may mislead the Bayesian network construction method, further contributing to the discovery of a sub-optimal net </li></ul>
12. Simpson’s Paradox (Pearl 2000) <ul><li>Overall E (recovered) E (not recov.) Total Recov. Rate </li></ul><ul><li>Drug (C) 20 20 40 50% </li></ul><ul><li>No Drug ( C) 16 24 40 40% </li></ul><ul><li>Total 36 44 80 </li></ul><ul><li>Males E (recovered) E (not recov.) Total Recov. Rate </li></ul><ul><li>Drug (C) 18 12 30 60% </li></ul><ul><li>No Drug ( C) 7 3 10 70% </li></ul><ul><li>Total 25 15 40 </li></ul><ul><li>Females E (recovered) E (not recov.) Total Recov. Rate </li></ul><ul><li>Drug (C) 2 8 10 20% </li></ul><ul><li>No Drug ( C) 9 21 30 30% </li></ul><ul><li>Total 11 29 40 </li></ul>
13. Simpson’s Paradox as a Surprising Pattern <ul><li>Event C (“cause”) increases the probability of event E (“effect”) in a given population but, at the same time, decreases the probability of E in every subpopulation </li></ul><ul><li>No paradox in terms of probability theory, it looks a “paradox” under a causal interpretation </li></ul><ul><ul><li>Gender is a confounder variable in the previous example </li></ul></ul><ul><li>Although Simpson’s paradox is known by statisticians , occurrences of the paradox are surprising to users </li></ul><ul><li>There are algorithms that systematically find instances of the paradox in data and rank them in decreasing order of surprisingness (Fabris & Freitas 2006) </li></ul>
14. The proposed method for integrating Bayesian networks and Simpson’s paradox <ul><li>Basic Idea: </li></ul><ul><ul><li>In a Bayesian network, the dependence denoted by edge C E can be spurious, i.e., due to a confounding variable F </li></ul></ul><ul><ul><li>(for the previously discussed reasons) </li></ul></ul><ul><li>Two approaches exploring this basic idea </li></ul>
15. First Approach: paradox detection before network construction <ul><li>First, run an algorithm that detects occurrences of Simpson’s paradox in data (Fabris & Freitas 2006) </li></ul><ul><ul><li>Produces a paradox list PL </li></ul></ul><ul><li>Modify Bayesian network construction algorithms to take into account this list, biasing the algorithms against including network edges involving the paradox </li></ul><ul><li>Consider a potential dependence represented by the edge C E, where C is apparent cause of effect E </li></ul><ul><ul><li>If variables C, E are associated in an occurrence of Simpson’s paradox in PL, the algorithm is biased against including edge C E in the network </li></ul></ul>
16. Consider a greedy algorithm that starts with an empty network and adds one edge to the network at a time, guided by a scoring function FOR EACH candidate edge A B compute the score of the network if A B is added to the network penalize score if there is an occurrence of the paradox in list PL involving pair of variables A, B SELECT edge with highest score and add it to the network proposed extension
17. The same basic kind of extension can be applied to an Estimation of Distribution Algorithm – EDA is a population-based evolutionary algorithm – It evaluates a complete candidate solution (network) at once FOR EACH candidate solution in the population compute the score of the network represented by the candidate solution penalize score in proportion to the number of paradox occurrences in list PL that are associated with direct dependences A B in the network proposed extension
18. Second Approach: paradox detection after network construction <ul><li>First, construct a Bayesian network from data </li></ul><ul><li>Use the network to “prune” the search space for the Simpson’s paradox detection algorithm </li></ul><ul><li>The algorithm will focus its search on the pairs of variables for which there is a direct dependence (i.e., an edge A B ) in the Bayesian network </li></ul><ul><li>For each pair of such variables, the algorithm will try to find a third variable that acts as a confounder between those two variables </li></ul>
19. Bayesian net variables considered by Simpson’s paradox detection algorithm, considering the Bayesian net Cause Effect Is there a counfounder? A C ? B C ? C D ? A B C D A paradox occurrence involving the above pairs of cause and effect variables would be even more surprising to the user, due to the structure of the network
20. Limitation of the proposed integration method <ul><li>It is possible that the data does not contain any occurrence of Simpson’s paradox </li></ul><ul><ul><li>In this case the usefulness of the method is limited </li></ul></ul><ul><li>Even if the algorithm does not find any paradox occurrence, this result is to some extent useful: </li></ul><ul><ul><li>it gives us increased confidence that the dependences represented in the network are true dependences, rather than spurious ones </li></ul></ul><ul><ul><li>This additional test complements (rather than replaces) conventional methods for evaluating Bayesian networks </li></ul></ul>
21. Conclusions <ul><li>We proposed a method for integrating two very different kinds of algorithm in data mining </li></ul><ul><ul><li>Algorithms for constructing Bayesian networks </li></ul></ul><ul><ul><ul><li>Discover potentially causal, more useful patterns </li></ul></ul></ul><ul><ul><li>Algorithms for detecting Simpson’s paradox </li></ul></ul><ul><ul><ul><li>Discover surprising patterns, potentially more useful </li></ul></ul></ul><ul><li>Hopefully, combining the “best of both worlds”, increasing the chance of discovering patterns useful for intelligent decision making by the user </li></ul><ul><li>Future research: computational implementation of the proposed method and analysis of results </li></ul>
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